Convergence of
PageRank and HITS
Algorithms
Victor Boyarshinov
Eric Anderson
12/5/02
Outline
Algorithms
Convergence
Graph data and a bad graph
Results
PageRank Algorithm
initialize ranks R0
while (not converged)
for each vertex i
end
end
HITS Algorithm
initialize authority and hub weights, x0 and
y0
while (not converged)
for each vertex i
end
end
Convergence
Many sensible options:
Maximum change between iterations
Sum of changes between iterations
Change of top q% of weights
Choice: sum of changes
Performance of PageRank
Converges in O(log(n)) iterations on
expander graphs
Motivation: propagation depends on
diameter
Iterations are expensive
Constant in order could have a large
influence
Graph Data
Synthetic data
Erdös-Rényi model
Chose to keep mean out-degree
constant
Standard mean out-degree: 10
Size on the order of thousands of
vertices
Bad Graph
Constructed from two random graphs
of equal size
Single link and backlink from one
cluster to the other
Idea: bottleneck slows propagation
Hypothesis: iterations will grow like
diameter: twice that of each cluster
Check: O(2*log(n/2)) iterations?
Some PageRank Results
Size Iterations Size Iterations
1000 4 1000 4
2000 5 2000 5
4000 5 4000 5
8000 5 8000 5
16000 6 16000 6
Summary of PageRank
results
Hypothesis failed completely
Changing edge probability changes
iterations, but not comparative
performance
Seemingly impossible to stump
PageRank
Conclusion
PageRank is stable
HITS is stable
Nearly doubling the diameter has no
noticeable effect on convergence